Perron-Frobenius theory and max-algebraic combinatorics of nonnegative matrices

极大代数是一个快速发展的数学分支，具有解决数学、科学和工程中一类非线性问题的潜力，当算术加法被极大运算取代，算术乘法被加法取代时，这些问题可以被赋予线性问题的形式。除了像处理线性问题一样处理非线性问题的优势之外，极大代数技术使我们在许多情况下能够有效地描述所有解的集合，从而根据特定的标准选择最佳解。它还提供了一类组合或组合优化问题的代数编码。尽管极大代数的基础是50年前在英格兰中心的第一篇开创性论文中创建的，但主要是在2000年之后，我们才在世界范围内的一些研究中心（如巴黎、伯克利、圣地亚哥、代尔夫特、麦迪逊和莫斯科）看到了这一领域的显著扩张。如今，它渗透到数学的一系列领域，从代数拓扑、泛函分析、线性代数和几何，到非线性、离散和随机优化以及数学生物学。部分或全部致力于最大代数的会议、小型专题讨论会、讲习班和其他活动的数量正在增加。出版了多部研究专著，其中三本是2005年以后出版的。应用既有理论（例如离散事件动态系统，控制理论和优化），也有实际（荷兰铁路网的分析）。随着近年来最大代数的显著发展和最新的研究成果，利用最近发展起来的强大的最大代数组合技术来加强最大代数与传统线性代数之间的相互作用似乎是一个合适的时机。这意味着例如，在半环中发展Perron-Frobenius理论，发展最大代数张量理论，解决平均收益博弈和最大代数矩阵方程。这将对矩阵的一系列性质的理解产生直接的影响，这些性质在其他数学领域，在物理学，计算机科学，工程，生物学和其他类似于传统线性代数的领域中得到应用。例如，它将使研究人员能够解决系统的最大代数方程，帮助分析复杂的信息技术系统，通过使用最大代数而不是传统的模型，找到一个稳定的政权系统与最大线性动力学，模型和解决固态物理问题，或在某些类型的调度问题。为了支持这个项目，也为了帮助解决这些挑战，PI将把这项研究与由他担任主席的伦敦数学学会资助的英国热带数学工作组的工作联系起来。该小组的研究会议每年在沃里克、曼彻斯特和伯明翰举办三次，来自英国多所大学的30多名同事参加了会议。PI将与一些国际领先的最大代数中心建立合作网络和战略伙伴关系，以进一步推动该领域的发展。预计由于该项目，PI将获得支持，于2014年或2015年在伯明翰组织一次热带数学国际会议，并在未来建立一个热带数学中心（CTM），该中心将有几个资助的研究项目。CTM将组织国际研究研讨会和会议，为工业合作伙伴以及专业的本科和研究生课程提供专业知识。它将与曼彻斯特大学的CICADA研究小组以及巴黎综合理工学院现有的类似中心密切合作。

Max-algebra is a rapidly evolving branch of mathematics with potential to solve a class of non-linear problems in mathematics, science and engineering that can be given the form of linear problems, when arithmetical addition is replaced by the operation of maximum and arithmetical multiplication is replaced by addition. Besides the advantage of dealing with non-linear problems as if they were linear, the techniques of max-algebra enable us in many cases to efficiently describe the set of all solutions and thus to choose a best one with respect to a specified criterion. It also provides an algebraic encoding of a class of combinatorial or combinatorial optimisation problems.Although the foundations of max-algebra were created in the first pioneering papers produced in the heart of England 50 years ago, it is mainly after 2000 that we see a remarkable expansion of this area in a number of research centres worldwide (e.g. Paris, Berkeley, San Diego, Delft, Madison and Moscow). Nowadays it penetrates a range of areas of mathematics from algebraic topology, functional analysis, linear algebra and geometry, to non-linear, discrete and stochastic optimisation and mathematical biology. The number of conferences, mini-symposia, workshops and other events devoted partly or wholly to max-algebra is increasing. A number of research monographs have been published, three of them since 2005. Applications are both theoretical (for instance in discrete-event dynamic systems, control theory and optimisation) and practical (analysis of the Dutch railway network). Following the recent remarkable expansion of max-algebra and latest research findings, it seems to be the right time to use the recently developed powerful combinatorial techniques of max-algebra to strengthen the interplay between max-algebra and conventional linear algebra. This means for instance to develop the Perron-Frobenius theory in semirings, develop the theory of max-algebraic tensors, solve the mean-payoff games and max-algebraic matrix equations. This will have an immediate impact on the understanding of a range of properties of matrices which find applications in other areas of mathematics, in physics, computer science, engineering, biology and elsewhere in a way similar to that of conventional linear algebra. For instance it will enable researchers to solve systems of max-algebraic equations, help to analyse complex systems of information technology by using a max-algebraic rather than traditional model, find a steady regime in systems with max-linear dynamics, model and solve problems arising in solid state physics, or in certain types of scheduling problems.To feed into this project and also to help to address the challenges, the PI will link this research with the work of the existing UK working group in tropical mathematics funded by the London Mathematical Society, which he chairs. Research meetings of this group are organised three times a year in Warwick, Manchester and Birmingham and are attended by more than 30 colleagues from a number of UK universities. The PI will form collaborative networks and strategic partnership with a number of internationally leading centres in max-algebra to further advance the field. It is expected that as a consequence of this project the PI will obtain support to organise an international conference on tropical mathematics at Birmingham in 2014 or 2015 and in the future to create a centre for tropical mathematics (CTM), which will have several funded research projects. CTM will organise international research workshops and conferences, provide expertise for industrial partners and for specialised undergraduate and postgraduate courses. It will closely cooperate with the research group CICADA at the University of Manchester and with the existing similar centre at the Ecole Polytechnique in Paris.